Hybrid modeling Annemie Bogaerts Department of Chemistry, University of Antwerp (UA), Belgium...

Hybrid modelingHybrid modeling

Annemie Bogaerts

Department of Chemistry, University of Antwerp (UA),

BelgiumAnnemie.Bogaerts@ua.ac.be

College “Computational Plasma Physics” TU/e, 6/2/04

1. Introduction Glow discharge + applications

2. Overview of models for GD plasma Advantage of hybrid models

3. Hybrid modeling network for GD plasma Submodels + coupling

4. Particle-in-cell modeling for magnetron

5. Modeling network for laser ablation

Contents

-

V

+

cathode dark spacenegative glow anode zone

cathode anode

1. Introduction

Glow discharge:

e.g.: cell dimensions: cm3

argon gas (p ~ 1 Torr)V ~ 1 kV, I ~ mA

-

Ar0

Ar+e-

e- + Ar0 Ar* + e-

Ar+ + 2 e-

Ar+

M0 + e-

+ Ar*m

+ Ar+

Ar0 + e- + h

M+ , M*

Processes in the glow discharge:

anode

cathode

1. Introduction (continued)s

pu

tte

rin

g

second.electronemission

1. Introduction (continued)

Applications of glow discharges:• Analytical spectrometry (source for MS, OES)

• Semiconductor industry (deposition, etching)

• Materials technology (deposition)

• Lasers

• Light sources

• Flat plasma display panels

• Environmental, biomedical applications

• …

Aim of our work:Better understanding to improve results Modeling

Principle:

Simple analytical formulas,

valid for specific range of conditions

2. Overview of models for GD plasmaA. Analytical model

Advantage: Simple, fast

Disadvantage: Approximation, limited validity

Principle:

Moment equations of Boltzmann equation:

Conservation of mass, momentum, energy

(or: drift-diffusion approximation)

2. Overview of models for GD plasmaB. Fluid model

Advantage:

Simple, fast

Coupling to Poisson equation for self-consistent E-field

Disadvantage:

Approximation (therm. equilibrium)

Principle:

~ Fluid model

Conservation equations for excited species

(balance of production + loss processes)

2. Overview of models for GD plasmaC. Collisional-radiative model

Advantage: Simple, fast

Disadvantage: /

Principle:

Full solution of Boltzmann transport equation

(terms with E-gain, E-loss)

2. Overview of models for GD plasmaD. Boltzmann model

Advantage: Accounts for non-equilibrium behavior

Disadvantage: Mathematically complex

Principle:

• Treats particles on lowest microscopic level

• For every particle – during successive time-steps:

* trajectory by Newton’s laws

* collisions by random numbers

2. Overview of models for GD plasmaE. Monte Carlo model

Advantage:

Accurate (non-equilibrium behavior) + simple

Disadvantage:

Long calculation time + not self-consistent

Principle:

• Similar to MC model (Newtons’ laws, random numbers)

• But at every time-step: calculation of electric field

from positions of charged particles (Poisson equation)

2. Overview of models for GD plasmaF. Particle-in-cell / Monte Carlo model

Advantage: Accurate + self-consistent

Disadvantage: Even longer calculation time

Which model should I use ?

Every model has advantages +

disadvantages…

Solution: use combination

of models !!

Principle: Because some species = fluid-like, others = particle-like:

Combination of above models, e.g.:

• Monte Carlo for fast (non-equilibrium) species

• Fluid model for slow (equilibrium) species + E-field

2. Overview of models for GD plasmaG. Hybrid model

Advantage:

• Combines the advantages + avoids the disadvantages

• Accurate + self-consistent + reduced calculation time

Disadvantage: /

Combination of different models for various species

Species: Model used:

Ar0 atoms no model (assumed thermalized)

or: heat conduction equation

electrons MC for fast electrons fluid for slow electrons

Ar+ ions fluid (with electrons + Poisson) MC in sheath

Ar0f fast atoms MC in sheath

Ar* excited atoms collisional-radiative model

Cu0 atoms thermalization after sputtering: MC

Cu*, Cu+(*), Cu++ collisional-radiative model

Cu+ ions MC in sheath

3. Hybrid modeling network for GD plasma

A. Heat transfer equation for Ar atoms

Gas temperature = f (power input in Ar gas):

P

r

Tr

rr

1

z

T g

2

g2

Power: calculated in MC models:

• elastic collisions of Ar+ ions, Ar0f atoms, Cu atoms,

and electrons with Ar atoms

• reflection of Ar+ ions and Ar0f atoms at cell walls

nAr = p / (k Tg)

During successive time-steps: Follow all electrons:

• Trajectory by Newton’s laws:

• Probability of collision:

Compare with RN (0 - 1): If Prob < RN => no collision If Prob > RN => collision

B. Monte Carlo model for fast electrons

tm

)sin(Eqvvt

m2

)sin(Eqtvyy

tm

)cos(Eqvvt

m2

)cos(Eqtvxx

tm

Eqvvt

m2

Eqtvzz

radyy

2rady0

radxx

2radx0

axzz

2axz0

00

00

00

))E(n(sexp1obPr collcoll

• Kind of collision: partial collision prob. + compare with RN

0 1

Pexcit Pioniz Pelast …

• New energy + direction: scattering formulas + RN:

Repeat until electrons: (Ep + Ek) < Ethreshold (typically in NG, where E-field weak)

transfer to slow electron group

B. Monte Carlo model for fast electrons (cont.)

)RN1(81

RN21cos

Continuity equations for ions + electrons:

RAr+ and Re: from MC model

Transport equations for ions + electrons (drift-diffusion):

Poisson’s equation:

C. Fluid model for slow electrons + Ar+ ions

eee

ArArAr Rj

t

n|Rj

t

n

eeeeeArArArArArnDEnj|nDEnj

VE|0nne

V eAr0

2

Only in CDS (where strong E-field) !

During successive time-steps: Follow all ions + atoms:

• Trajectory by Newton’s laws

• Probability of collision, • Kind of collision, defined by RN’s• New energy + direction after coll.

Repeat this until:• Ar+ ions bombard cathode• Ar0

f atoms: E < Ethermal

D. MC model for Ar+ ions + Ar0f atoms

E. Collisional-radiative model for Ar* atoms

lossprod2*Ar

2

*Ar*Ar

*Ar*Ar RR

z

nD

r

nr

rr

1D

t

n

65 Ar levels (individual or group of levels)

For each level: continuity equation (balance equation):

Production + loss processes for each level: * electron, Ar ion + Ar atom impact excitation, de-excitation, ionization* electron-ion three-body recombination, radiative recombination* radiative decay* Hornbeck-Molnar associative ionization (for Ar* levels with E > 14.7 eV)

Additional processes for 4s levels:* Ar* - Ar* collisions -> (associative) ionization* Penning ionization of sputtered atoms* three-body collisions with Ar atoms* radiation trapping of resonant radiation

Energy level scheme for Ar* CR model

E (eV) E (1000 cm-1) jc = 3/2 jc = 1/2

s p d f,… s’ p’ d’ f’,…

130

125

120

115

110

105

100

95

90

1

2

3 4

5

(4s)

(3p6)

(4s’)

11.5

16.0

15.5

15.0

14.5

14.0

13.5

13.0

12.5

12.0

0 0

6

7

10 8

9

11

(4p)

(4p’)

(3d) (5s)

(5p)

(5s’) (3d’)

(5p’)

12

13 16

18

14

17 15

19

(4d’)

(4d)

(6s’)

(6s) 20

21 22

23 (6p)

(7s)

(8s) (9s)

(7p) (8p) (9p)

(5d)

(6d) (7d)

(8d) (9d)

25

27 29 31

33 35 37 39 41 43 45 47 49

51

57 - 65

53 55

(7s’)

(8s’) (9s’)

(6p’)

(7p’)

(8p’)

(9p’)

(5d’)

(6d’) (7d’)

(8d’) (9d’)

24

26 28 30

32 34 36 38 40

42 44 46 48

50 52 54 56 - 64 Ar+ (3p5) 2P1/2

Ar+ (3p5) 2P3/2

F. Cu0: sputtering, thermalization

Sputtering:

• EDF of Ar+, Ar0, Cu+ (from MC models)

• Sputtering yield Y(E): empirical formula

Result = Jsput

Thermalization: Monte Carlo model• cfr. electron Monte Carlo model

Result = FT

FT * Jsput: source term for Cu0 in next model

G. Cu0, Cu*, Cu+, Cu+*: Collisional-radiative model

8 Cu0, 7 Cu+ levels, 1 Cu++ level:

For each level: balance equation:

Production + loss processes:

• Cu0: FT * Jsput

• electron, atom impact excitation, de-excitation • electron, atom impact ionization, recombination• radiative decay• Penning ionization, asymmetric charge transfer

Transport:

diffusion for Cu0, diffusion + drift for Cu+ and Cu++

Cu,lossCu,prodCuCu RRJ.t

n

Energy level scheme for

Cu* CR model

E (eV)

0

1

5

4

3

2

15

11

10

9

8

7

6

28

22

21

20

19

18

17

16

1

5

4

3

2

8 7 6

11

10

9

Cu0 3d10 4s 2S1/2

3d9 4s2 2D5/2

3d9 4s2 2D3/2

3d10 4p 2P1/2

3d10 4p 2P3/2

3d9 4s 4p 4P,4D,4F

3d10 5p 2P1/2,3/2

3d10 5s 2S1/2

3d10 4d 2D3/2,5/2

3d9 4s 4p 2P,2D,2F

3d10 6p 3d10 6s 3d9 4s 4p 2P,2D,2F 3d10 5d

Cu+ 3d10 1S0

16

15

14

13

12 3d9 4p 3P2

3d9 4p 1P1

3d9 4p 3P1,0,3F,3D,1F3,

1D2

3d9 5s 3D1,2,3 , 1D2

3d9 5p 3d9 4d

3d8 4s 4p 5D,5G,5F

3d8 4s2 3F,1D,3P,1G

3d9 4s 3D1,2,3 3d9 4s 3D1,2,3

Cu++ 3d9

3d9 4s 1D2

H. Cu+ ions in CDS: Monte Carlo model

• cfr. Ar+ ion Monte Carlo model

• Cu+ ions: important for sputtering !

I. Coupling of the models

Ar+/e- fluid model

e- MC model Ar+/Ar0f MC model

Ar* CR model

Cu MC model

Cu/Cu+ CR model

Cu+ MC model

arbitrary RAr+ , Re,slow

Eax and Erad dc(r), jAr+,dc(r), jAr+,0(r)

Re,ion,Ar

Ri,ion,Ar and Ra,ion,Ar

new RAr+ and Re,slow

convergence ? no yes

e- MC model: - Re,exc,Ar, Re,exc,met, Re,ion,met: for Ar*m model - Re,ion,Cu : for Cu model Ar+/Ar0

f MC model: - Ri,exc,Ar, Ra,exc,Ar : for Ar*m model - fAr+(0,E), fAr(0,E) : for Cu model Ar+/e- fluid model: - nAr+ , ne,slow : for Ar*m model - nAr+ , V : for Cu model

nAr,met

FT

Rion,Cu , jCu+,dc(r) nCu Ar*m model: - nAr,met : for Re,exc,met and Re,ion,met

- prod, loss: for nAr+ , ne,slow Cu model: - nCu : for Re,ion,Cu - nCu+ : for E, V - asymm. CT : for nAr+ - ioniz.terms: for ne,slow

convergence ? no

yes

final solution

fCu+(0,E)

Detailed coupling

• Output:

* Electric field: Eax, Erad

* Interface CDS – NG: dc (r)

* Ar+ ion flux entering CDS: jAr+, dc (r)

* Ar+ ion flux at cathode: jAr+,0 (r)

Start: Ar+ - e- fluid model:

• Input: arbitrary creation rates: RAr+, Re

• Electric field (Eax, Erad): to calculate trajectories of

electrons and ions (Newton’s laws)

• Interface CDS – NG + Ar+ flux entering CDS: jAr+, dc (r):

to define Ar+ ions starting in Ar+ MC model

• Ar+ ion flux at cathode: jAr+,0 (r): to define electron flux

starting at cathode (secondary electron emission):

je,0 (r) = - jAr+,0 (r)

This output = input in e-, Ar+, Ar0f MC models:

• Electron MC model (1)

Input: from fluid model

Output: electron imp. ionization rate = creation of Ar+ ions

• Ar0f MC model (1):

Input: from Ar+ MC model

Output: ionization collisions: creation of e-, Ar+

• Ar+ MC model (1):

Input: from fluid model + e- MC model (creation of Ar+)

Output: * Creation of Ar0f (elastic collisions)

* Creation of e- (ionization collisions)

Coupling between e-, Ar+, Ar0f MC models:

• Ar+ MC model (2):

Extra input: from Ar0f MC model (creation of Ar+)

Same output: ( = more creation of Ar0f and e- )

Coupling between e-, Ar+, Ar0f MC models (cont):

Etc… until CVG reached (creation of e- constant)

• Ar+ MC model (3)

• Ar0f MC model (3)

• …

• Ar0f MC model (2):

Extra input: from Ar+ MC model

Same output: (= more creation of e-, Ar+ )

• Electron MC model (2)

Input: from fluid + Ar+, Ar0f MC model (creation of e-)

Same output: (= more creation of Ar+ ions)

Coupling between e-, Ar+, Ar0f MC models (cont):

Etc… until CVG reached:

total creation of Ar+ ions and e- constant

(typically: after 2-3 iterations)

• Again: Ar+ / Ar0f MC models

• Again: e- MC model (3)

• …

Coupling betw. e-, Ar+, Ar0f MC models (summary):

e- MC model

Ar+ MC model Ar0f MC model

e- MC model

Creation of Ar+

Creation of Ar+

Creation of Ar0f

Creation of e- Creation of Ar+

• Output of Ar+ - e- fluid model (= same):

* Electric field: Eax, Erad

* dc (r) and jAr+, dc (r)

* Ar+ ion flux at cathode: jAr+,0 (r)

= Again input in MC models

Coupling between MC models + fluid model:

• Global output of MC models:

Creation rates of Ar+, electrons: RAr+, Re

= Input in Ar+ - e- fluid model (2)

Etc… until CVG reached (E-field, jAr+,0 constant)

(typically after 5-10 iterations)

Coupling between MC + fluid models (Summary):

e- - Ar+ fluid model

e-, Ar+, Ar0f MC models

CVG ?

Arbitrary creation of Ar+ , e-

Creation of Ar+ , e-

• E-field

• dc(r), jAr+,dc(r)

• jAr+,0 (r)

Yes

No

Similar hybrid MC - fluid models for GD plasmas:

• L.C. Pitchford and J.-P. Boeuf (Toulouse)

• Z. Donko (Budapest)

Remark:

This is “core” of hybrid model

When Ar density ≠ constant, but calculated in heat transfer model Additional model in loop:

Initial e- - Ar+ fluid model

e-, Ar+, Ar0f MC models

CVG ?

Arbitrary creation of Ar+ , e- + constant nAr

• Creation of Ar+ , e- ( Fluid model)• Power input in Ar gas ( Heat transf.model)

E-field, dc(r), jAr+,dc(r), jAr+,0 (r)

Yes

No Ar heat transfer model

New nAr (function of position)

Coupling between MC + fluid models = strongest coupling

Hybrid MC – fluid model determines: • Structure of GD (CDS, NG)• Electric field distribution

• Ar+ ion and electron densities• Electrical characteristics (I-V-p)• …

Other models: • Do not affect electrical structure of GD• Are important for specific applications (e.g., spectrochemistry)

Other models: more loosely coupled

Results of the MC + fluid models used as input in the other models:

From e- MC model:

• Electron impact ionization, excitation, de-excitation

rates of Ar ( used as populating + depopulating

terms in Ar CR model)

• Electron impact ionization, excitation, de-excitation

rates of Cu ( used as populating + depopulating

terms in Cu CR model)

Results of the MC + fluid models used as input in the other models (cont):

From Ar+ and Ar0f MC models:

• Ar+ and Ar0f impact ionization, excitation, de-excitation

rates of Ar ( used as populating + depopulating

terms in Ar CR model)

• Ar+ and Ar0f flux energy distributions at cathode

( used to calculate flux of sputtered Cu atoms)

Results of the MC + fluid models used as input in the other models (cont):

From Ar+ - e- fluid model:

• densities of Ar+ ions and electrons

( used in some populating + depopulating terms

in Ar CR model and Cu CR model, i.e., recombination)

• density of Ar+ ions ( used in Cu CR model,

for the rate of asymmetric charge transfer ionization:

Ar+ + Cu0 Ar0 + Cu+)

• E-field distribution ( used in Cu CR model,

to calculate transport of Cu+ ions by migration)

Using all this input:

the Ar CR model + 3 Cu models

(i.e., Cu MC thermalization model,

Cu CR model and Cu+ MC model)

are solved in coupled way

(1) Ar CR model:

• Input: cfr. above, + constant Cu atom density

• Output (among others): Ar metastable atom density

( used in Cu CR model, for the rate of

Penning ionization: Ar*m + Cu0 Ar0 + Cu+ + e-)

Coupling of the 3 Cu models

(i.e., Cu MC thermalization model,

Cu CR model and Cu+ MC model)

(1) MC model for Cu thermalization after sputtering:

• Input: * Ar+ and Ar0f flux EDFs (from MC models)

* Ar density (constant or from heat transf.model)

• Output: Thermalization profile

( used in Cu CR model, for initial distribution of

Cu atoms (product: FT * Jsput))

Coupling of the 3 Cu models (cont)

(2) Cu CR model:

• Input: * from MC + fluid models (cfr. above)

* FT*Jsput from Cu thermalization MC model)

• Output (among others):

* Flux of Cu+ ions entering CDS from NG

* Creation rate of Cu+ ions in CDS

( both used in Cu+ MC model in CDS)

Coupling of the 3 Cu models (cont)

(3) Cu+ MC model in CDS:

• Input: from Cu CR model (cfr. above)

• Output (among others):

Cu+ ion flux EDF at cathode

( used to calculate flux of sputtered Cu atoms)

Coupling of the 3 Cu models (cont)

With this extra output of Cu+ MC model (i.e., Cu+ EDF):

Again calculation of :

• Sputtering flux (empirical formula)

• MC model for Cu thermalization

• Cu CR model

• Cu+ MC model in CDS

Etc… until cvg reached (i.e., when sputter flux = constant)

(typically after 2 – 3 iterations)

Coupling to the Ar CR model

Output of the 3 Cu models (among others): Cu0 density

( used in Ar CR model, to calculate

rate of Penning ionization: Cu0 + Ar*m Cu+ + Ar0 + e-)

Etc… until cvg reached

(i.e., when Cu0 and Cu+ density constant)

(typically after 2 iterations)

Output of Ar CR model (Ar metastable atom density):

again used as input in the 3 Cu models

Summary: Coupling of Ar CR model and the 3 Cu models

Ar CR model

Cu sputtering flux (emp.formula)MC model for Cu thermalization

Output from Ar+ / e- MC + fluid models

• Cu+ flux entering CDS

• Cu+ creation rate in CDS

Ar*m density

Cu CR model

FT * Jsput

Cu+ MC model in CDS

Cu+ EDF at cathode

Cu0 density(coupling back

until CVG)

Results of the Ar CR model + 3 Cu models

used as input in the Ar/e- MC + fluid models:

From Ar CR model:

• Populations of Ar metastable + other excited levels

( used to recalculate the e-, Ar+ and Ar0f impact

ionization, excitation, de-excitation rates of Ar

in the MC models)

• Some production + loss terms of Ar excited levels

( can influence densities of Ar+ and e- in fluid model)

Results of the Ar CR model + 3 Cu models

used as input in the Ar/e- MC + fluid models:

From the 3 Cu models:

• Cu atom density

( used to recalculate the e- impact ionization,

excitation, de-excitation rate of Cu in MC model)

• Cu+ ion density

( can influence E-field distribution in fluid model)

• Ionization rates of Cu (EI, PI, aCT) ( can influence

densities of Ar+ and e- in fluid model)

However:

Results of the Ar CR model + 3 Cu models

do not really affect the calculations

in the Ar/e- MC + fluid models

Hence:

Coupling back not really necessary

at typical operating conditions

Summary

Ar+/e- fluid model

e- MC model Ar+/Ar0f MC model

Ar* CR model

Cu MC model

Cu/Cu+ CR model

Cu+ MC model

arbitrary RAr+ , Re,slow

Eax and Erad dc(r), jAr+,dc(r), jAr+,0(r)

Re,ion,Ar

Ri,ion,Ar and Ra,ion,Ar

new RAr+ and Re,slow

convergence ? no yes

e- MC model: - Re,exc,Ar, Re,exc,met, Re,ion,met: for Ar*m model - Re,ion,Cu : for Cu model Ar+/Ar0

f MC model: - Ri,exc,Ar, Ra,exc,Ar : for Ar*m model - fAr+(0,E), fAr(0,E) : for Cu model Ar+/e- fluid model: - nAr+ , ne,slow : for Ar*m model - nAr+ , V : for Cu model

nAr,met

FT

Rion,Cu , jCu+,dc(r) nCu Ar*m model: - nAr,met : for Re,exc,met and Re,ion,met

- prod, loss: for nAr+ , ne,slow Cu model: - nCu : for Re,ion,Cu - nCu+ : for E, V - asymm. CT : for nAr+ - ioniz.terms: for ne,slow

convergence ? no

yes

final solution

fCu+(0,E)

Input data for the modeling network

• Electrical data: Voltage, pressure, gas temperature

Electrical current = calculated self-consistently

• Reactor geometry: (e.g., cylinder: length, diameter)

• Gas (mixture)

• Cross sections, rate coefficients,

transport coefficients,…

All other quantities: calculated self-consistently

Typical calculation results

General calculation results:* Electrical characteristics (current, voltage, pressure)

* Electric field and potential distribution

* Densities, fluxes, energies of the plasma species

* Information about collisions in the plasma

Results of analytical importance:* Crater profiles, erosion rates at the cathode

* Optical emission intensities

* Effect of cell geometry, operating conditions

Electrical characteristics:For given V,p,T: calculate current

Calculated: Measured:

Illustrations of some results

Densities:Argon metastable density (1000 V, 1 Torr, 1.8 mA):

Calculated: Measured (LIF):

z (cm )

r (c

m)

0E+000

1E+009

2E+009

3E+009

4E+009

5E+009

1E+010

3E+010

1E+011

5E+011

1E+012

c m - 3

0

0.5

1.0

1.5

2.0

-0.5

-1.0

-1.5

-2.00 0.5 1.0 1.5 2.0

z (cm )

r (c

m)

0E+000

1E+007

5E+007

1E+008

2E+008

5E+008

1E+009

2E+009

3E+009

4E+009

c m - 3

2.0

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

-2.00 0.5 1.0 1.5 2.0

Densities:Sputtered tantalum atom density (1000 V, 1 Torr, 1.8 mA):

Calculated: Measured (LIF):

z (cm )

r (c

m)

0E+000

1E+010

5E+010

1E+011

2E+011

5E+011

1E+012

2E+012

3E+012

c m - 3

2.0

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

-2.00 0.5 1.0 1.5 2.0

z (cm )

r (c

m)

0.0E+000

1.0E+010

1.0E+011

2.0E+011

4.0E+011

6.0E+011

8.0E+011

1.0E+012

1.3E+012

1.6E+012

c m - 3

2.0

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

-2.00 0.5 1.0 1.5 2.0

Densities:Tantalum ion density (1000 V, 1 Torr, 1.8 mA):

Calculated: Measured (LIF):

z (cm )

r (m

)

0E+000

5E+008

1E+009

2E+009

4E+009

6E+009

8E+009

9E+009

c m - 3

2.0

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

-2.00 0.5 1.0 1.5 2.0

z (cm )

r (c

m)

0E+000

1E+009

5E+009

1E+010

2E+010

3E+010

4E+010

5E+010

6E+010

8E+010

c m - 3

2.0

1.5

1.0

0.5

0

-0.5

-1.0

-1.5

-2.00 0.5 1.0 1.5 2.0

Energies:Electron energy distribution (1000 V, 0.56 Torr, 3 mA):

Energies:Argon ion energy distribution (1000 V, 0.56 Torr, 3 mA):

Calculated: Measured (MS):

Energies:Copper ion energy distribution (1000 V, 0.56 Torr, 3 mA):

Calculated: Measured (MS):

Information about sputtering at the cathode:Crater profile after 45 min. sputt. (1000 V, 0.56 Torr, 3 mA):

Calculated:

Measured:

0

50000

100000

150000

200000

250000

300000

350000

400000

300 400 500 600 700 800 900 1000

(nm)

Inte

ns

ity

(a

.u.)

696.

5470

6.72

727.

2973

8.40

750.

3975

1.47

763.

5177

2.38

794.

8280

0.62

801.

4881

0.37

811.

53

826.

4584

0.82

842.

4685

2.14

866.

79

912.

30

922.

4593

5.42

965.

7897

8.45

0.E+00

5.E+14

1.E+15

2.E+15

2.E+15

3.E+15

3.E+15

4.E+15

4.E+15

5.E+15

5.E+15

300 400 500 600 700 800 900 1000

(nm)

Inte

ns

ity

(a

.u.)

415.

8642

0.07

434.

52

603.

39

604.

49

696.

5470

6.72 73

8.4

750.

3875

1.47

763.

5177

2.38

- 7

72.4

279

4.82

800.

62 -

801

.48

811.

5382

6.45

840.

8284

2.47

852.

1486

2.29

866.

79

912.

392

2.46 96

5.78

978.

45

Optical emission intensities:Ar(I) spectrumCalculated:

Measured:

4. Particle-in-cell / Monte Carlo model for magnetron discharge

Hybrid MC – fluid model for magnetron discharge:

Very complicated (B) – approximations necessary

Only suitable for certain B/n values

Fluid model: Effect of magnetic field

Initial try: Rigourous description for electron flux Assumptions:* 1D magnetic field (Bx)

* Ar+ ions do not feel magnetic field

y

nD

z

nDEnEn

1

1

y

nD

z

nDEnEn

1

1

eex

ee

yeex

zee

2

2x

ez

ee

eex

yee

zeex

2

2x

ey

very complicated

Fluid model: Effect of magnetic field (contd)

Therefore: Approximation: Effect of magnetic field in electron transport coefficients:

0 02 2 2 2

1 1 and D ,

1 1B B

eD qB m

Note:

* If based on cross sections : / ~ 500

* However, Bradley: / ~ 7.7 4.2 (max. 25)

We use as fitting parameter (physically realistic results)

This gives: / ~ 15 (depending on position)

= electron gyro-frequency = average electron momentum transfer collision frequency

Limitations of the hybrid model

* Strong B/n: time of electron confinement

negative space charge, negative plasma potential

= Only observed exper. at much higher B (3000 G)

* Alternative: Transport coefficients calculated

from Boltzmann equation (swarm parameters)

However: only done for 1D + constant B

In practice: Hybrid MC-fluid approach:

only useful for certain B/n (e.g. p > 10 mTorr, B < 200 G)

Real plasma particles: replaced by superparticles

1 superparticle = W real particles (W = weight, e.g. 2x107)

Integration of equation of motion, moving

particles

Fi vi’ xi

Particle loss/gain at the boundaries (emission,

absorption)

Integration of Poisson’s equations on the grid

()i (E)i

Weighting

(x,v)i ()I

(interpol. charges to grid)

Weighting

(E,B)j Fi

(interpolate field to particles)

MCCollision

?

No

ΔtYes Postcoll.

velocities vi

’ vi

Δtelec = 1 ps

Δtion = 10Δtelec

PiC-MC model for magnetron discharge

5. Modeling network for Laser Ablation

laser

target

Heat conduction T melting evaporation

•Plume expansion• Laser absorption

by plasma

laser

target

Heat conduction T melting evaporation

Physical picture:

Typical operating conditions:

• Pulsed lasers (Nd:YAG or excimer):

ns, ps, fs - laser pulse (10 ns fwhm)

• UV – IR wavelengths ( = 266 nm)

• Laser intensity ~ 107-1010 W/cm2 (109 W/cm2)

• Laser beam diameter ~ 100 m

• Target = metals, non-metals (Cu)

• Expansion in vacuum – 1 atm background gas (He, Ar, air)

Applications of Laser Ablation:

• Pulsed laser deposition

• Nanoparticle manufacturing

• Micromachining

• Surgery

• Spectrochemistry (MALDI, LIBS, LA-ICP-MS)

Different processes to describe in a model:

• Laser-solid interaction: heating, melting, vaporization

• Evaporated plume expansion (in vacuum)

• Plasma formation in the plume

• Laser – plasma interaction (plasma shielding)

• (Plume expansion in background gas: interactions)

• (Nanoparticle formation in the plume)

Hybrid modeling network

• 1D heat conduction equation:

• Laser intensity:

• Melting: data for molten Cu

• Vaporization

vapor density, velocity, temperature

= Input in next model

1. Target heating, melting, vaporization

)t,x(ICx

)t,x(T

Cxt

)t,x(T

pp

1xexp)t(I)t,x(I 0

Conservation equations:

(mass density, momentum, energy)

Godunov method (shock waves)

2. Expansion of evaporated plume

radlaserIB

22

2

I2

vpEv

x2

vE

t

vpxt

v

x

v

t

• Saha equations for ionization degree:

Ratio Cu+/Cu0:

Ratio Cu2+/Cu+:

• Conservation of matter:

• Conservation of charge:

• Internal energy density:

3. Plasma formation

Tk

IPexp

h

Tkm2

n

1

x

xx

Tk

IPexp

h

Tkm2

n

1

x

xx

2

2/3

2e

vap1i

2ie

1

2/3

2e

vap0

1ie

e2i1i

2i1i0

xx2x

1xxx

2i211i1e x)IPIP(xIPTk)x1(

2

3

mE

Inverse Bremsstrahlung:

Heating term in energy conservation eq.

4. Laser beam absorption in plasma

2i2

21i2

1

2/1

ee4

e36

ie,IB

0ene,IB

nZnZTkm3

2

mch3

ne4

Tk

chexp1

nnQTk

chexp1

1) Laser-target interaction: evaporation

Vapor dens, veloc, temp plume expansion

(boundary conditions for Euler equations)

5. Coupling of the different parts

2) Absorption of laser beam in plasma

• Energy gain in plume (term in Euler equation)

• Plasma shielding before reaching target

(Intensity at target << Laser intensity)

Therefore:

To describe entire picture of GD plasma (or LA):

Combination of different models desirable

General conclusion

Different plasma species (or aspects):

require different modeling approach

Hybrid model